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Definition

A cogenerator in a categoryCC is an objectSS such that the functorhS=C(−,S):Cop→Seth_S = C(-,S) : C^{\mathrm{op}} \to \mathrm{Set} is faithful. This means that for any pair g1,g2∈C(X,Y)g_1,g_2\in C(X,Y), if they are indistinguishable by morphisms to SS in the sense that

One often extends this notion to a cogenerating family of objects, which is a (usually small) set 𝒮={Sa,a∈A}\mathcal{S} = \lbrace S_a, a\in A\rbrace of objects in CC such that the family C(−,Sa)C(-,S_a) is jointly faithful. This means that for any pair g1,g2∈C(X,Y)g_1,g_2\in C(X,Y), if they are indistinguishable by morphisms to 𝒮\mathcal{S} in the sense that

Proof

Let CC be a set of generators for the topos; as usual, let Ω\Omega be the subobject classifier. We claim that a product

∏c∈CΩc\prod_{c \in C} \Omega^c

is a cogenerator. For suppose f,g:X→→Yf, g \colon X \stackrel{\to}{\to} Y are distinct morphisms. The contravariant power object functor Ω−\Omega^- is faithful (a familiar fact, since it is monadic), so that Ωf,Ωg:ΩY→→ΩX\Omega^f, \Omega^g: \Omega^Y \stackrel{\to}{\to} \Omega^X are distinct. Since the objects cc form a generating set, there is some h:c→ΩYh \colon c \to \Omega^Y such that the composites

are distinct. For any other c′∈Cc' \in C, we may uniformly define Y→Ωc′Y \to \Omega^{c'} to be the map classifying the maximal subobject of c′×Yc' \times Y, so that these maps together with h˜\tilde{h} collectively induce a map

Y→∏c∈CΩcY \to \prod_{c \in C} \Omega^c

that yields distinct results when composed with ff and gg. This proves the claim.

The object ∏Ωc\prod \Omega^c is injective because already Ω\Omega is injective (see Mac Lane-Moerdijk, IV.10), and it is a general fact that in a cartesian closed category (or more generally a closed monoidal category), an exponential (or internal Hom) XYX^Y whose base XX is injective is also injective, and products of injective objects are injective.